I live with my girlfriend, a molecular biologist, and her passion for topics like the brain and cells often leads to fascinating (if sometimes very scientific) conversations. So when I started researching the concept of Cellular Automata, I saw a wonderful subject that combined both of our fields in an unexpected way. If there's one overall takeaway from this post, it's that math truly is everywhere!
So what is it?
in a nutshell; Cellular Automata are mathematical models that simulate complex systems from surprisingly simple rules.
Developed initially by John von Neumann and later popularized by Stephen Wolfram. Ever heard of Wolfram|Alpha - the incredible math engine used by countless students and experts alike? The research company behind it was founded by this guy.
His findings in 1983 showed incredibly complex and unpredictable behavior, and describing a very powerful tool for modeling real-world phenomena.
Basic principles
Representation
Cellular Automata are typically represented by a grid of cells. Each cell can be in one of a finite number of states. For example, in the simplest form of Cellular Automata, each cell is either "on" or "off" (binary states). The grid can be one-dimensional, two-dimensional, or even higher-dimensional, though two-dimensional grids are the most commonly studied.
States and Rules
The rules of Cellular Automata determine how the state of each cell evolves over time. These rules are usually based on the states of neighboring cells. For example, a rule might specify that a cell turns "on" if exactly two of its neighbors are "on" and turns "off" otherwise. Despite the simplicity of these rules, they can produce insanely complex and varied behavior.
Game Of Life
The Game of Life, developed by mathematician John Conway, is perhaps the most famous example of Cellular Automata . In this two-dimensional grid, each cell is either alive or dead. The rules for the Game of Life are:
- Any live cell with fewer than two live neighbors dies (underpopulation).
- Any live cell with two or three live neighbors lives on to the next generation.
- Any live cell with more than three live neighbors dies (overpopulation).
- Any dead cell with exactly three live neighbors becomes a live cell (reproduction).
My rendition of the phenomena, built in Javascript
Sand Simulation
This simulation models the behavior of falling sand grains, showing how simple rules can create realistic-looking phenomena. Each cell represents a grain of sand, air, or a solid surface, and the rules dictate how sand grains move in response to gravity and collisions.
My rendition in Javascript and HTML canvas
So, how does it actually work?:
From these examples, we can draw out three phases of life for a cell:
1. Initialization
The starting conditions, or initial state, of the Cellular Automata grid are crucial. These conditions can be randomly generated, or they can be set up in a specific pattern to observe how different configurations evolve over time.
2. Rules
The rules of Cellular Automata define the "neighborhood" of a cell (the set of cells that influence its state) and the state transitions (how a cell's state changes based on its neighbors). In the Game of Life, for example, the neighborhood consists of the eight surrounding cells, and the state transitions are defined by the rules mentioned earlier.
3. Iteration
Cellular Automata evolve over discrete time steps, with each cell updating its state simultaneously based on the rules. This process is repeated for many iterations, often resulting in intricate and unexpected patterns.
We'll see soon this show up in weird ways!
Emergent Patterns and Behavior:
One of the most fascinating aspects of Cellular Automata is this weird emergence of complex behavior from simple rules. Where larger patterns and structures arise out of local interactions.
Self-replication and Growth
Some Cellular Automata rules can lead to self-replicating structures, which are patterns that produce copies of themselves. These structures can be used to model biological processes like cell division and growth.
Oscillators and Gliders
In 2D Cellular Automata like the Game of Life, certain patterns repeat infinitely, called oscillators:
This is incredible when thinking in the aspect of biological life - these cells are in harmony, and in a way 'perfect'.
Others travel across the grid, called gliders:
Showing interesting applications in how human cells might be travelling through our insides.
These patterns aren't just visually interesting but also demonstrate how complex the end result can actually get - and how we're moving in the direction of what Stephen Wolfram described as 'Scientific Philosophy'.
Stability and Chaos
Furthermore Cellular Automata can exhibit a range of behaviors from stable (where the grid reaches a steady state like the Oscillator mentioned above) to chaotic (where patterns change unpredictably over time). The study of these behaviors helps scientists understand the transition between order and chaos in complex systems.
Applications of Cellular Automata:
Now here's where it gets interesting. Until now we've simply described how simple rules gives complex results, yada yada - but remember how i told you that the iterations showed up in very weird ways?
Simulations of natural phenomena
Phew, well here we go. Cellular Automata can even be used to model biological processes like the development of organisms, pattern formation, and the spread of all kinds of diseases. Take these patterns on seashells for example.
Like, man, this is legit just Conway's Game of Life from above - I don't know about you, but I'm pretty blown away.
Seeing the rules we specified earlier of how each cell moves through different iterations, now in real life, is nothing short of incredible.
To keep it 'Computer Sciency' and not just about Lizards, Cellular Automata have actually also been proposed for public-key cryptography, which makes sense given their complexity. The evolution of Cellular Automata patterns over numerous iterations is highly unpredictable, making it extremely difficult to reverse-engineer or predict previous states. This inherent unpredictability provides a robust foundation for encryption - but if I'm not wrong, it's still just a theory and haven't been implemented as of yet.
Simply put, this stupid little thing finds its place in an incredible array of contexts—from complex computer systems to small lizards whose scales provide camouflage, helping them hide from predators while they search for food.
As research in Cellular Automata continues, our understanding of these emergent behavior improves, and who knows where the next application will find itself.
Conclusion
Cellular Automata provide a unique lens through which we can explore the complexity that arises from simple rules. As we delve deeper into their properties and behaviors, we continue to uncover new insights that could revolutionize various scientific and technological fields.
And it's also just freaking cool!
Further Reading and Resources:
I've only scratched the surface, there's all kinds of definitions like Wolfram's class system etc. - so a whole world is available to read about if this simple introduction was interesting.
Here's are some influential papers and books:
- Stephen Wolfram's "A New Kind of Science"
- Toffoli and Margolus: "Cellular Automata Machines: A New Environment for Modeling"
Experiment with my sand simulation
Experiment with my version of 'Game of Life'
I'll end on a beautiful and relevant quote from the great Stephen Wolfram.
...All the wonders of our universe can in effect be captured by simple rules, yet ... there can be no way to know all the consequences of these rules, except in effect just to watch and see how they unfold.
Thank you for reading.
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